Existence, Localization and Multiplicity of Positive Solutions to $\phi$-Laplace Equations and Systems

Abstract

The paper presents new existence, localization and multiplicity results for positive solutions of ordinary differential equations or systems of the form $(\phi(u'))' + f(t, u) = 0$, where $\phi : (-a, a) \to (-b, b)$, $0 \lt a, b \leq \infty$, is some homeomorphism such that $\phi(0) = 0$. Our approach is based on Krasnosel'skiĭ type compression-expansion arguments and on a weak Harnack type inequality for positive supersolutions of the operator $(\phi(u'))'$. In the case of the systems, the localization of solutions is obtained in a component-wise manner. The theory applies in particular to equations and systems with $p$-Laplacian, bounded or singular homeomorphisms.